High frequency behavior of the Leray transform: model hypersurfaces and projective duality

TL;DR

The paper develops a sharp L^2 theory for the Leray transform on the unbounded model hypersurfaces $M_\gamma$ in ${\mathbb C}^2$, including exact norms, high-frequency behavior, and spectral descriptions of $L^*L$ and related operators. It uses a Fourier-analytic decomposition arising from $S^1$-symmetry to obtain precise sub-Leray kernels $\mathbf L_k$, their symbols $C_\sigma(\gamma,k)$, and the limiting high-frequency norm, tying these to projective invariants. A projective-duality framework is built via the dual hypersurface $M_{\gamma^*}$ with $\gamma^* = \gamma/(\gamma-1)$, along with Fefferman and a specially chosen “preferred” measure to define invariant Hardy spaces on $M_\gamma$ and their duals on $M_{\gamma^*}$. The authors establish sharp Cauchy–Schwarz inequalities in the Leray pairing, construct dual Hardy spaces, and formulate a robust pairing between these spaces via a pairing measure $\nu^{A_\gamma}$. Overall, the work elucidates the high-frequency behavior of the Leray transform on unbounded domains and strengthens the connection between function theory and projective geometry through invariant Hardy spaces and duality.

Abstract

The Leray transform $\bf{L}$ is studied on a family $M_γ$ of unbounded hypersurfaces in two complex dimensions. For a large class of measures, we obtain necessary and sufficient conditions for the $L^2$-boundedness of $\bf{L}$, along with an exact spectral description of $\bf{L}^*\bf{L}$. This yields both the norm and high-frequency norm of $\bf{L}$, the latter giving an affirmative answer to an unbounded analogue of an open conjecture relating the essential norm of $\bf{L}$ to a projective invariant on a bounded hypersurface. $\bf{L}$ is also shown to play a central role in bridging the function theoretic and projective geometric notions of duality. Our work leads to the construction of projectively invariant Hardy spaces on the $M_γ$, along with the realization of their duals as invariant Hardy spaces on the dual hypersurfaces.

Theorems & Definitions (95)

• Theorem 1.3
• Theorem 1.4
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.11
• Theorem 1.13
• Conjecture 2.16
• Remark 2.20
• Theorem 3.2